The distance from a vertex u to a vertex υ in a connected graph G is the length of a shortest u-υ path in G. The distance of a vertex υ of G is the sum of the distances from υ to the vertices of G. For a vertex υ in 2-edge-connected graph G, we define the edge-deleted distance of υ as the maximum distance of υ in G-e over all edges e of G. A vertex is an edge-deleted distance stable vertex if the difference between its edge-deleted distance and distance is 1. A 2-edge-connected graph G is an edge-deleted distance stable graph if each vertex of G is an edge-deleted distance stable vertex. In this paper, we investigate the edge-deleted distance of vertices and describe properties of edge-deleted distance stable graphs.
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